
Integral equations of this type occur in [1]}, [2]}. By
considering \(W+\int _0^. \hat{a}_r \Phi _0 dr\)  as a Brownian motion with
respect to some probability measure which is equivalent to
\(\mathbb {P}\) , we can apply the results from Section 6.4 in [2]}
(in particular see Theorem 6.3 and its proof there). We obtain that
(REF ) has a unique solution given by
\(\hat{W}_t& = W_t+\Phi _0\int _0^t \hat{a}_s ds-\int _{0}^t\int _{0}^s \hat{\kappa }_{s,r} \left(dW_r+\Phi _0a_r dr\right) ds \nonumber \\&=W_t-\int _{0}^t\int _{0}^s \hat{\kappa }_{s,r} dW_r ds+\Phi _0\left(\int _{0}^t\hat{a}_s ds-\int _{0}^t\int _{0}^s \hat{\kappa }_{s,r}\hat{a}_r dr ds\right)\) 
